Green of Greece - Logic and Mathematics

Tuesday, July 11, 2017

The strategy of this proof is more
elaborate than the preceding post but retains the key idea of analyzing the
curve y = xnin terms of
the three variables a, b and c in the form of a diagram.The new proof introduces a model of
Fermat’s Conjecture based on the
implications of the anti-conjecture, the aim being to
identify inherent contradictions.

The difficulty of the problem lies in the consistency of the equations derived from the hypothesis that the
conjecture is false, the result being a series of tautologies that lead
nowhere. Although the equations that arise appear to imply the irrationality of a key
variable, it is usually the case that this condition cannot be established
conclusively. In such cases the strategy is to reject the irrationality and
follow up the implications of that decision. The situation is reminiscent of
the Escher picture shown above, where everything seems to be consistent in two
dimensions but must surely be false in a three dimensional world.

By extracting equations from the diagram, rather than just
manipulating consistent equations derived algebraically,
the inconsistencies in the assumptions are revealed and can be incorporated into a proof. This works because the geometric
picture contradicts the equations derived from the defining equation an + bn = cn.

Friday, November 6, 2015

My latest proof of Fermat's Conjecture is available for download here: Fermat Paper .A slide show video can be seen here FLT video The introduction and a few passages are listed below, subject to formatting limitations, together with the conclusions and explanatory diagrams.

The main idea is that a unique tangent to the curvexn is determined by the variables a and c. The perpendicular drawn from the point where the tangent meets the cuve cuts the x-axis at K and it is the properties of K that determine the structure of the proof. The significance of this parametric variable is explained below in the passage about Pythagorean Triples.

A key property of the curve, when n > 2, is that equal decrements to the left of K correspond to unequal increments to the right. This is because the differential coefficient of the curve is

nKn-1
being non-linear but when n = 2, the differential coefficient is a linear function.

The proof strategy is to show that the variables a and c cannot both be integers when K is an integer. However, it is pretty clear thatK is never an integer, so two further proofs are required under the assumptions that K is a rational fraction and K is an irrational number.

When K is an integer it is impossible for the secant to pass through integer points in the plane. When K is a rational fraction, this is also the case. WhenK is irrational then the secant can pass through integer points but the variable b cannot be rational. This situation constitutes the proof of FLT.

The mathematics involved in the proof is elementary and would certainly have been within the lexicon of Pierre de Fermat. An image of his 1621 translation of Diophantus whose margin could not hold Fermat's “marvelous proof”is shown below.

Introduction

The
aim of this paper is to establish a more direct proof of Fermat’s Last Theorem than the proof published by Andrew Wiles in The Annals of Mathematics 142 (1995). The
main idea is that the three terms an,
bnand cn all lie on the curve xn which facilitates the
construction of diagrams showing the relation between these terms and several
auxiliary variables used in the proof. The two diagrams used are shown in the
annex.

Fermat’s Conjecture

Fermat’s
Conjecture is: There are no natural numbers a,
b, c, n such that an + bn
= cn when n > 2.

FLT
may be expressed formally as: NSabcn∈ℕ
an + bn = cn

subject
to the following conditions:

(1)a,b,c
and n are distinct natural numbersabcnℕ

(2)a,b and c have no common factorsabc NCF

(3)n is greater than
2.n>2

(4)
a useful conventiona<b<c

(5)
n is a prime number

Parity limitations

Condition
(2) implies that a, b and c cannot all be even numbers, otherwise they would have the common
factor 2. Furthermore, if any two
variables are even numbers then they have the common factor 2. Consequently, two of the factors must
be odd and the third even. It will be shown later that b must be an odd variable given condition (3) so either a or c
is even.

Analysis of K

This
special characteristic explains why Pythagorean triples are possible because a and c always lie symmetrically about K. It will be shown below that this is not the case when n > 2 so the secant can never simultaneously
intersect integer values of (a, an)
and (c, cn) when K∈ℕ.

The
variables a and c can be defined in terms of K
as follows:

a = K – q
and c = K + p so that c – a = p + q, where p and q are deviations on either side of K.

A secant may be formed by moving the tangent an integer distance to the left of K.

When
n =2 an equal deviation occurs to the
right of K so that p = q. The underlying reason is that the
differential coefficient of x2 is
the linear function 2x. When n
> 2it is intuitively evident
that an integer move to the left will result in a lesser and possibly
fractional deviation to the right where p
< q.

Summary and conclusion

(1)
If K is an integer then g is not an integer so either a is not an integer or c is not an integer.

(2)
If K is a rational fraction then g cannot be an integer so either a is not an integer or

c
is not an integer

(3)
If K is an irrational number then
either n is irrational or b is irrational.

In
arriving at these conclusions the negative hypothesisSabcn∈ℕ an + bn = cn
was assumed together with the conditions (1) through
(5) applicable to FLT. The three conclusions above prove that the negative
hypothesis under the given conditions is inconsistent, consequently FLT